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Given the Hohenberg-Kohn theorem, a given ground-state density uniquely defines all ground state properties, because in principle the external potential $V_{ext}$ can be understood as a functional of the ground-state density $n_0(r)$.

Now the Schrödinger equation gives me - again, in principle, the ground-state wave function $\Psi_0(r)$, so if I knew the functional $V_{ext}[n(r)]$, I could go from a ground-state density to the ground-state wavefunction.

This might seem counter-intuitive at first, because the density itself seems to not contain any phase information, but Hohenberg-Kohn says that, in fact, all the information is there (just very well hidden, and not readily extracted).

The question is now: Is the missing phase information contained more in the density or more in the mysterious functional? If it's the latter, I would expect that DFT using approximations to the exchange-correlation functional should break down for phenomena that rely somewhat strongly on (local) phase, such as Anderson localization or the Aharonov-Bohm effect?

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    $\begingroup$ Your intuition is in fact correct. The usual theorems have a loop-hole in terms of thermodynamics. Specifically, although for a finite system one could use any basis (and so a free-electron one is as good as any), in a properly infinite system there could be a phase transition, which makes certain states impossible to reach starting from a de-localised basis. Indeed, strongly correlated phases tend to cause problems with DFT. $\endgroup$ – genneth May 30 '12 at 3:03
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When you go from the wavefunction to the density, you are throwing away more than just the phase: you are losing information about electron-electron correlation. One could think of this information as being in some sense "stored" in the exchange-correlation functional, which is a very complicated beast whose exact form is unknown.

In Kohn-Sham DFT, the one-electron wavefunctions can have a phase. (This contrasts with "orbital-free" DFT, in which you only consider the electron density). So in Kohn-Sham DFT, broadly speaking there is good reason to expect that common approximations to the XC functional (such as the local-density approximation) in principle could capture the physics for systems in which phase was important. Where the approximations run in to trouble is situations in which correlation is important (e.g. van der Waals dispersion interactions, or transition metals).

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